Complete second direction in main lemma

main.tex

@@ -248,14 +248,14 @@ Suppose that the following are satisfied:
 
 \noindent
 Then we have the following:
-\begin{itemize}
+\begin{enumerate}
 	\item The pseudo-wall is left of $u$'s vertical characteristic line
 		(if this is a real wall then $v$ is being semistabilized by an object with
 		Chern character $u$, not $-u$)
-	\item $\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is left of $v$'s vertical
-		characteristic line
+	\item $\beta(P)<\mu(u)<\mu(v)$, i.e., $u$'s vertical characteristic line is
+		positioned between $P$ and $v$'s vertical characteristic line
 	\item $\chern_2^{P}(u)>0$
-\end{itemize}
+\end{enumerate}
 Furthermore, only the last two of these consequences are sufficient to recover
 all of the suppositions above.
 \end{lemma}
@@ -368,8 +368,6 @@ def correct_hyperbola_intersection_plot():
   p.ymax(coords_range[1][2])
   p.ymin(coords_range[1][1])
   p.axes_labels([r"$\beta$", r"$\alpha$"])
-
-  
   return p
 \end{sagesilent}
 
@@ -400,8 +398,20 @@ $\beta$-axis at $\beta=\mu(u)$ and $\beta=\mu(v)$ respectively.
 We must have $\mu(u)<\mu(v)$, that is, the vertical characteristic line for $u$
 is to the left of the one for $v$ (consequence 2).
 Finally, the fact that it is the left branch of the hyperbola for $u$ implies
-consequence 1.
-
+consequence 1 and $\beta{P}<\mu(u)$.
+
+
+Conversely, suppose that the consequences 2 and 3 are satisfied.
+Consequence 2 implies that the assymptote for the left branch of the
+characteristic hyperbola for $u$ is to the left of the one for $v$.
+Consequence 3, along with $\beta{P}<\mu(u)$, implies that $P$ must be in the
+region left of the left branch of the characteristic hyperbola for $u$.
+These two facts imply that the left branch of $u$'s hyperbola is to the right of
+that of $v$'s at $\alpha=\alpha(P)$, but crosses to the left side as
+$\alpha \to +\infty$. This implies suppositions 1 and 2, and that the
+characteristic curves for $u$ and $v$ must be in the configuration illustrated
+in Fig \ref{fig:correct-hyperbol-intersection}.
+Recalling consequence 3 finally confirms supposition 3.
 \end{proof}
 
 \begin{sagesilent}